Security implications of enabling IPv6 on my VPS?

My experience is that most applications will Just Work (at most, you might need to say "yes, listen on IPv6 too", but most apps do that by default).

As far as security goes, the most important thing to remember is that anything you've done to restrict the flow of IPv4 packets needs to be recreated in IPv6. Most people boil this down to "make sure your firewall is blocking IPv6 as well as IPv4", but there can be other things than firewalls doing the blocking (hosts.allow/deny, ACLs in applications, that sort of thing), and you need to make sure all of that is working.

If you do any monitoring, you'll want to ensure that you're monitoring both IPv4 and IPv6 versions, to ensure that you're not providing degraded service to IPv6 clients, and make sure any log analysis tools will understand IPv6 addresses.

Nothing else comes immediately to mind. Enjoy the brand new Internet.